JAMB 2003 · UME · Q34

Plane mensuration
Circle geometry

$XYZ$ is a circle with centre $O$ and radius 7 cm, and $\angle XZY = 45^\circ$ . Find the area of the shaded segment cut off by $XY$ . $\left[\pi = \frac{22}{7}\right]$

Worked solution (try it first)

The angle at the centre is twice the angle at the circumference, so $\angle XOY = 2 \times 45^\circ = 90^\circ$ .
The quarter-circle sector is $\frac14 \times \frac{22}{7} \times 7^2 = 38.5\text{ cm}^2$ .
The right-angled triangle $XOY$ is $\frac12 \times 7 \times 7 = 24.5\text{ cm}^2$ .
Segment = sector − triangle: $38.5 - 24.5 = 14\text{ cm}^2$ , option A.

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